Dont laugh, we truly enjoy Lincoln Park Zoo. One of the highlights was seeing the Polar bear kept swimming up towards the glass wall, so up close to the viewers on the other side of the wall... He kept doing this for good 10 minutes back and forth back and forth. And the tigers doing the same thing... While we enjoy watching animals in the wild like a self-drive safari in South Africa, seeing well-cared for animals in a zoo is also a great pleasure.
Lincoln Park also has a Conservatory that is nice to visit.
Actually the several museums, such as Field of Natural History while inside the city are quite good, too. If your daughter hasn't been, it is a good place to spend a day (there are other museums in the vicinity).
Would the Botanical Garden be too hot in Summer? It is a vast place and not much shades as far as I remember.